Inverse Mills Ratio - Uses

Uses

Use of the inverse Mills ratio is often motivated by the following property of the truncated normal distribution. If X is a random variable having a normal distribution with mean μ and variance σ2, then

\begin{align} & \operatorname{E} = \mu + \sigma \frac {\phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}, \\ & \operatorname{E} = \mu + \sigma \frac {-\phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}{\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}, \end{align}

where α is a constant, ϕ denotes the standard normal density function, and Φ is the standard normal cumulative distribution function. The two fractions are the inverse Mills ratios.

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